In general no. In the majority of languages, string literals are 'interned', which hashes them and makes their lookup much faster. In general, there may be some discrepancies between different javascript engines, but overall if they're implemented well (cough IE cough), it should be fairly equal. Especially since javascript engines are constantly being developed, this is (probably) an easy thing to optimize, and the situation will improve over time.
However, some engines also have limits on the length of strings that are interned. YMMV on different browsers. We can also see some insight from the jsperf test (linked in comments for the question). Firefox obviously does much more aggressive interning.
As for the types of characters, the string is treated as just a bunch bytes no matter the charset, so that probably won't matter either. Engines might optimize keys that can be used in dot notation but I don't have any evidence for that.